Plasma and Fusion Research: Regular Articles

Volume 11, 1402002 (2016)

A Calibration of Setting of Mach Probes by Observing GAM
Oscillations
Kimitaka ITOH1,2) , Kaijun ZHAO3) , Jiaqi DONG3) , Sanae-I. ITOH2,4) , Akihide FUJISAWA2,4) ,Shigeru INAGAKI2,4) , Makoto SASAKI2,4) , Yoshihiko NAGASHIMA2,4) , Yusuke KOSUGA2,4,5) ,Jun CHENG3) and Tatsuya KOBAYASHI1)
1)

National Institute for Fusion Science, Toki 509-5292, Japan
Research Center for Plasma Turbulence, Kyushu University, Kasuga 816-8580, Japan
3)
Southwestern Institute of Physics, PO Box 432, Chengdu, People’s Republic of China4)
Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan5)
Institute for Advanced Study, Kyushu University, Fukuoka 812-8581, Japan2)(Received 10 December 2015 / Accepted 25 December 2015)
The influence of relative displacement of Mach probe (which is placed near the top of magnetic surface) on
the interference of signals is discussed. An error can arise in measured value of poloidal electric field. The Mach
number perturbation at the GAM frequency has an interference from the density perturbation. The interference
from the density perturbation can propagate to all of Mach number measurement. By observing the signals
associated with GAM oscillations, the error in setting the probe arrays can be detected. This result can be applied
to correct the positioning of probes.
c 2016 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: geodesic acoustic mode, probe measurement, Mach probe, probe alignment
DOI: 10.1585/pfr.11.1402002

1. Introduction
In order to study the transport of toroidal momentum,
fluctuation of plasma flow along the magnetic field linewas measured by use of Mach probes (See, e.g., [1–4]).
Mach probe was recently applied to the study of geodesic
acoustic mode (GAM) [5]. The measurement of transport
fluxes requires the phase relation between several quantities [6–8], so that the cross talk between the flow along thefield line and other flow components or the density perturbation might cause the systematic error in the study oftransport characteristics.
One of the origins of incorrect observation is an error of the setting of the probe with respect to the magnetic
field lines and/or magnetic surfaces [9]; If a Mach probe is
not completely parallel to the magnetic field line, the deduced fluctuation of Mach number can include influencesof perpendicular velocities. The influence of other quantities (like density perturbation) is possible to occur. We
here discuss a possible method to calibrate the position of
the Mach probe in the edge of tokamak plasmas. By observing the signals associated with GAM oscillations, theerror of placing the probe arrays can be detected. This result can be applied to calibrate the position of probes. The
calibration method depends on what must be measured correctly: e.g., the accurate measurement of parallel flow fordrift wave fluctuations may require diﬀerent principle ofcalibration from that for GAM oscillations. In this note
author’s e-mail: sasaki@riam.kyushu-u.ac.jp

we study the case where the flows in GAM oscillations aremeasured.
Possible applications of GAMs to advanced diagnostics were proposed. For instance, GAM spectroscopy has
been proposed to measure the isotope ratio of fuel ions[10]. A method to identify the plasma surface by observing
GAM oscillation was proposed [11]. This note discusses
an additional application of GAMs for detailed analysis ofdiagnostics.

2. Geometry
An example of an array of Langmuir probes is illustrated in Fig. 1. A pair of probes 9 and 10 (or those of 11and 12) constitutes a Mach probe [5]. Local coordinates
on the probe array are defined as is illustrated. The x- andy-coordinates are (nearly) in the radial and poloidal directions. The z-coordinate is designed to take the direction ofmagnetic field line so as to measure the plasma flow velocity along the field line. The setting of the probe arrayis illustrated in Fig. 2. The Fig. 2 (a) shows the poloidal
cross-section. The major radius of the probe position may
be diﬀerent from that of the center of the magnetic surface.
The Fig. 2 (b) shows the top view. The Mach probe may
not be tangent to the magnetic field line and to the magnetic
surface. Two tips of the Mach probe may be on diﬀerent
magnetic surface. For the illustration of the problem, wediscuss here in the case of a circular magnetic surface in ahigh-aspect-ratio tokamak.

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c 2016 The Japan Society of Plasma
Science and Nuclear Fusion ResearchPlasma and Fusion Research: Regular Articles

Volume 11, 1402002 (2016)

Fig. 1 Front-view and side view of probe array are illustrated. The local (x, y, z) coordinates on the probe are also denoted.

3. Flow of GAM Oscillation and Calibration
The GAM oscillation has a particular feature oftoroidal symmetry, and the associated electric field is almost in the minor radius direction. The eigenfunction isexplained in [12]. Ratio of the toroidal component of velocity to the poloidal one was given as −2q cos θ for low
frequency zonal flows, (where q is the safety factor and θ isthe poloidal angle,) and that for GAMs is q−1 cos θ† . The
eigenfunction of GAMs is given as⎛
⎞0⎜⎜⎜
⎟⎟⎟⎜
⎟⎟⎟
Er ⎜⎜⎜1⎟⎟⎟ .
V = − ⎜⎜⎜(1)⎟⎟B ⎜⎝⎜ 1cos θ ⎠⎟q
The electric field is mainly in the radial direction, and thoseprojected on the x-y coordinates are given as
E x = Er sin θ,
Ey = −Er cos θ.

(2)That is,θ = arctan −E x /Ey .

(3)

Thus one can estimate the shift of probe position R withrespect to the top of the magnetic surface R , by measuring
the ratio between E x and Ey on the probe arrays as
(4)
R − R = r cos arctan −E x /Ey .
Fig. 2 Poloidal cross-section of the magnetic surface and theprobe array, (a). The local coordinates indicate the direction of the probe. The top view of the probes (b).
The circle indicates the major radius R. Inclination ofthe probe (relative rotation angle of the probe around theprobe stem) is indicated by the angle δ.

In interpreting the data Ey in terms of ‘Eθ ’, two probes
are assumed to be on the same magnetic surface. If two† It is well known that the impact, which induces a poloidal flow V0(with no toroidal flow), results in the low frequency zonal flow with theamplitude of (1 + 2q2 )−1 V0 after the decay of GAM oscillations, as isexplained in [12]. This is because the eigenfunction of GAM is given asEq. (1) and the initial poloidal flow (with no toroidal flow) is a combination of zonal flow and GAM, the poloidal velocities of which are given as(1 + 2q2 )−1 V0 and q2 (1 + 2q2 )−1 V0 , respectively.

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Volume 11, 1402002 (2016)

probes (which measure the ‘poloidal’ electric field) are notparallel to the magnetic surface, a small portion (that isproportional to Er ) is included in the interpreted ‘Eθ ’. For
GAM, the ratio between Eθ and Er is known, and is zeroat top (θ = π/2). Thus, one can evaluate a small angle
between the two probes and the magnetic surface.
We next consider possible inclination of the Machprobe with respect to the magnetic field line. The Mach
number M is estimated by comparing the ion saturation
currents at two probes
M = 0.4 ln (I s1 /I s2 ),Combining Eq. (3) and Eq. (6a), one has
Vtor /Vθ ∼ q−1 cos arctan −E x /Ey .

(6a)n1 − n2 = −ikr Δñ,(8)where ñ is the amplitude of density perturbation of GAM
oscillation. Since the ion saturation current is proportional
to the product of density and ion sound speed, I s = nc s ,the ratio I s1 /I s2 is approximately evaluated as
I s1 /I s2 = 1 + (n1 − n2 ) /n0 = 1 − i (kr Δ) ñ/n0 .

(9)Here, n0 is the mean density, and we simply assume thatthe relative temperature fluctuation of GAM is smaller thanthat of density change at GAM frequency. Substituting
Eq. (9) into Eq. (5), one comes to the estimate that theprobe provides the signal
Mint = −0.4i (kr Δ) ñn−10 ,√eφ̃ñ= − 2kr ρ ssin θ.
n0Te(10a)(11)

Equation (10b) shows that the apparent value of M, combined with the density perturbation Eq. (11) and Eq. (3),gives an evaluation of the angle δ.
An evaluation of tolerance in the misalignments may
be deduced from Eq. (10). Combining Eq. (1) and Eq. (11),the Mach number for GAM eigenmode is given as
1 ñ cos θ.
MGAM = −i √q 2 n0 sin θ(12)From Eqs. (10b) and (12), the ratio of Mint (apparent valueby misalignment) to MGAM is given as√Mint= 0.4 2 (kr l) q sin δ sin θ.
MGAM(13)

Note these two quantities have the same phase. The condition, that the apparent contribution Mint by error in placing
is much smaller than the real value for GAM, is rewritten
as|sin δ|  √(7)

where l is the distance of two probes in a Mach probe. Because GAMs have a large wave number in the radial direction, kr , this distance Δ introduces a diﬀerence of fluctuating density at probes 1 and 2 (the probe 1 is chosen as one
with the smaller z position) as(10b)where the suﬃx ‘int’ indicates the contribution of interference. In this process of interference, apparent Mach
number Mint and the density perturbation have the phase
diﬀerence with 3π/2. Because the GAMs have short radial
wavelength (of the order of 10ρ s ), the coeﬃcient 0.4 (kr l)can be of the order unity. The relative density perturbation
is estimated as(6b)Thus, if the fluctuating Mach number at GAM frequency,
being coherent with potential perturbations of GAMs,
shows the value much larger than Eq. (6a), it suggests that
the Mach probe may not be tangent to the magnetic surface. One possibility of the measurement error is that theobtained value of Mach number Eq. (5) is influenced by thedensity perturbation. Consider the case where the Machprobe is inclined by an angle δ as is illustrated in Fig. 2 (b)(top view). The distance in between the two probes, which
is projected in the direction of the minor radius, Δ, is givenasΔ = l sin δ cos θ,

Mint = −0.4i (kr l) ñn−10 sin δ cos θ,(5)where I s j ( j = 1, 2) is the ion saturation current at probe j.
The GAM eigenmode (1) has small parallel velocity near
the top of the magnetic surface as
Vtor /Vθ ∼ q−1 cos θ.

or

2.5
2 (kr l) q |sin θ|.

(14)

4. Discussion on Mach Probe Measurement
In measuring the flow velocity that is parallel to themagnetic field line, it is assumed that two probes in a pair
for a Mach probe is on the same magnetic field line. In
other words, the vector between the two probes (the z-axis
in Fig. 1) must be (almost) 90-degree to the major-radiusdirection. If this angle is misaligned, the measurement ofMach number includes an error. In the case of GAM, thetoroidal mode number is zero, so that the error in the angle
between major-radius-direction and probes-vector alone,
does not cause an error when the probe is correctly placedat the top. If the deviation of the probe position from the
top of magnetic surface exists simultaneously, the density
perturbation at one probe tip is diﬀerent from the density
perturbation at the other tip. Thus, Mach number perturbation at GAM frequency has an interference from the density perturbation. The interference from the density perturbation can propagate to all of Mach number measurement.
This hypothesis can be tested as follows.

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Volume 11, 1402002 (2016)(a) The relative distance R−R can be confirmed by MHD
equilibrium reconstruction. One can further confirm
Eq. (3) by observing the change of ratio between Eθ
and Er at GAM frequency under the condition ofmoving the toroidal axis of plasma (for fixed probe
position).
(b) In this process (of moving magnetic axis), the changeof observed Mach number perturbation at GAM frequency is observed. One may also rotate the probe(so as to change the angle δ between the major-radiusdirection and the vector of two probes), and the Mach
number perturbation at GAM frequency is observed.
If the hypothesis applies, “the Mach number perturbation
at GAM frequency” is a unique indicator to confirm therelevance of measurements.
The argument in this note can be extended for the caseof drift waves. An order of magnitude estimate is briefly
discussed here. In drift waves, the perturbation of the ion
parallel flow velocity has an order of magnitude of v|| ∼k|| eφ/(mi ω) [13]. That is, the perturbation in Mach numberisMdrift ∼k|| c s eφ.
ω Te(15)

Putting an estimate ω ∼ kθ ρ s c s /Ln , where Ln is the density
gradient scale length,
Mdrift ∼k|| Ln eφ k|| Ln ñ∼.
kθ ρ s T ekθ ρ s n0(16)For the simplicity of the argument, we take the case when
the probe is at top of the magnetic surface (R = R ) and thatit has a small inclination angle δ. The poloidal wave number is finite for drift waves, and the diﬀerence of density
perturbation at the two tips of Mach probe isn1 − n2 = −ikθ l sin δ ñ.

(17)

From Eq. (10a), one sees that an error in the Mach number
is induced by this inclination as
Mint ∼ −0.4ikθ l sin δñ.
n0(18)

Thus, the condition that the error in the evaluation in thevelocity fluctuation is much smaller than the real fluctuation of velocity for drift waves is evaluated as|sin δ|  2.5 k|| Ln /kθ ρ s (kθ l)−1 .

(19a)

If one puts further estimate, k|| ∼ (Rq)−1 and kθ ρ s ∼ 1,Eq. (19a) takes a form
|sin δ|  2.5 (Ln /qR) (ρ s /l) .

(19b)

This condition is usually more stringent than that for thecase of GAM oscillations (Eq. (14)). It is also noted thatthe error in M fluctuation by the inclination has a phase
diﬀerence by the amount of π/2, compared to the density
perturbation (or potential perturbation). This phase diﬀerence must also be taken care of when one evaluates theturbulence-driven flux.

Acknowledgments

This note is an outcome from the joint-data-analysis
workshop (November 12 to November 20, 2015 atKyushu University), which was organized in the framework of the international collaboration programme ofRIAM Kyushu University. The study is partly supported by a Grant-in-Aid for Scientific Research of JSPS,
Japan (23244113, 15H02155, 15K14282, and 15H02335),
Asada Science Foundation, National Science Foundationof China (11375054), and National Magnetic Confinement
Fusion Science Program of China (2013GB111000, and2013GB112008).
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